Real-rooted atlas

A008288 → basis A013609 → x ↦ x+1

nonnegative coefficients

Proved for all n all roots real, in (−∞, −1]coefficients of one signHadamard kernelinterlacing (weak)

Recurrence

\(\left(-1 + n\right)\,P_{n}(x) = \left(18 + 6\,x - 11\,n - 2\,n x + 2\,n^{2}\right)\,P_{n-1}(x) + \left(55 + 51\,x + 10\,x^{2} - 14\,n - 12\,n x - 2\,n x^{2}\right)\,P_{n-1}'(x) + \left(21 + 32\,x + 15\,x^{2} + 2\,x^{3}\right)\,P_{n-1}''(x)\)
\(P_{1}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
111
2426
31918441
488124568276
5409764520152161861
619004422402417843843212546
7882724598280281668054569286484581
84100813311218222413617659904154882176128570216
9190513705912112840010152485616961954564172849922563844201
10885076368602667383207090768472934420714885953281081601126451225916286
114111843190130183912235647142256368144961945440070403841721728272128250881024174718721
121910260097103988222102472301562680269912064167119744729992962248281647820806686725529620481177893756
138874592949188389212380109681870127144188750161613402952966862666242551758086835379212858368161177612083240967940956141
144122915162474544590679632704811305423496127031223041017736777659788944642606683520843078656199711232336670723823616262144819253535205626
151915403851123759049263683219806066900751448828260953447391160483249009703872245225836809301094656266028492856464281686214656894976056524816384360916014461

Other recurrences

Image under the basis map A007318 · proved equal for all n (proof step)

\(P_{n}(x) = \left(3 + 2\,x\right)\,P_{n-2}(x) + \left(4 + 2\,x\right)\,P_{n-1}(x)\)
\(P_{1}(x) = 1\)
\(P_{2}(x) = 4 + 2\,x\)
Technical details
Node id
n-2da37fcf9fa60a2f · revision 0 · JSON
Definition source
derived:shift_x_plus_one
derived:basis_map_A007318 (equal by e-d11a8911209dff68)
Fingerprint
f6-b4e11046d64dfe4b

Observed on the stored rows ?

  • Degrees: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14 (affine: slope 1, intercept -1)
  • Palindromic rows: no
  • γ-positive (gamma_positive): no
  • Nonnegative coefficients (nonn): yes
  • Only rational roots (rr_by_factors: every nonzero row is a product of linear factors over ℚ): no

Matches ?

No OEIS matches.

Submission text ?

Atlas rows n = 1, …, ascending powers of x
%e Example
Triangle begins:
   1:      1;
   2:      4,       2;
   3:     19,      18,       4;
   4:     88,     124,      56,       8;
   5:    409,     764,     520,     152,      16;
   6:   1900,    4422,    4024,    1784,     384,      32;
   7:   8827,   24598,   28028,   16680,    5456,     928,     64;
   8:  41008,  133112,  182224,  136176,   59904,   15488,   2176,    128;
   9: 190513,  705912, 1128400, 1015248,  561696,  195456,  41728,   4992,   256;
  10: 885076, 3686026, 6738320, 7090768, 4729344, 2071488, 595328, 108160, 11264, 512;
%F Formula
The row polynomials satisfy (n-1)*P(n,x) = (2*n^2-11*n+18-(2*n-6)*x)*P(n-1,x) - (14*n-55+(12*n-51)*x+(2*n-10)*x^2)*d/dx P(n-1,x) + (21+32*x+15*x^2+2*x^3)*d^2/dx^2 P(n-1,x) for n >= 2, with P(1,x) = 1.
%t Mathematica
P[1, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[((2*n^2 - 11*n + 18 - (2*n - 6)*x)*P[n - 1, x] - (14*n - 55 + (12*n - 51)*x + (2*n - 10)*x^2)*D[P[n - 1, x], x] + (21 + 32*x + 15*x^2 + 2*x^3)*D[P[n - 1, x], {x, 2}])/(n - 1)]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N)); v[1] = 1; for(n = 2, N, my(i = n); v[i] = ((2*n^2 - 11*n + 18 - (2*n - 6)*x)*v[i - 1] - (14*n - 55 + (12*n - 51)*x + (2*n - 10)*x^2)*deriv(v[i - 1], x) + (21 + 32*x + 15*x^2 + 2*x^3)*deriv(deriv(v[i - 1], x), x))/(n - 1)); v};
row(n, p) = {my(c = Vecrev(p)); c};
T(N) = {my(v = P(N)); concat(vector(#v, i, row(i, v[i])))};
T(10)

polytool c48629374597

Properties

Proof chains ? JSON

Same fingerprint ?

References

  1. [LW07] Lily L. Liu and Yi Wang. A unified approach to polynomial sequences with only real zeros. Advances in Applied Mathematics, 38(4):542–560, May 2007. doi:10.1016/j.aam.2006.02.003